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Theorem prmlem2 15770
Description: Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than 5↑2 = 25. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to 29↑2 = 841, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 15786).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Hypotheses
Ref Expression
prmlem2.n 𝑁 ∈ ℕ
prmlem2.lt 𝑁 < 841
prmlem2.gt 1 < 𝑁
prmlem2.2 ¬ 2 ∥ 𝑁
prmlem2.3 ¬ 3 ∥ 𝑁
prmlem2.5 ¬ 5 ∥ 𝑁
prmlem2.7 ¬ 7 ∥ 𝑁
prmlem2.11 ¬ 11 ∥ 𝑁
prmlem2.13 ¬ 13 ∥ 𝑁
prmlem2.17 ¬ 17 ∥ 𝑁
prmlem2.19 ¬ 19 ∥ 𝑁
prmlem2.23 ¬ 23 ∥ 𝑁
Assertion
Ref Expression
prmlem2 𝑁 ∈ ℙ

Proof of Theorem prmlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prmlem2.n . 2 𝑁 ∈ ℕ
2 prmlem2.gt . 2 1 < 𝑁
3 prmlem2.2 . 2 ¬ 2 ∥ 𝑁
4 prmlem2.3 . 2 ¬ 3 ∥ 𝑁
5 eluzelre 11658 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (ℤ29) → 𝑥 ∈ ℝ)
65resqcld 12991 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (ℤ29) → (𝑥↑2) ∈ ℝ)
7 eluzle 11660 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (ℤ29) → 29 ≤ 𝑥)
8 2nn0 11269 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℕ0
9 9nn0 11276 . . . . . . . . . . . . . . . . . . . . . . 23 9 ∈ ℕ0
108, 9deccl 11472 . . . . . . . . . . . . . . . . . . . . . 22 29 ∈ ℕ0
1110nn0rei 11263 . . . . . . . . . . . . . . . . . . . . 21 29 ∈ ℝ
1210nn0ge0i 11280 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ 29
13 le2sq2 12895 . . . . . . . . . . . . . . . . . . . . 21 (((29 ∈ ℝ ∧ 0 ≤ 29) ∧ (𝑥 ∈ ℝ ∧ 29 ≤ 𝑥)) → (29↑2) ≤ (𝑥↑2))
1411, 12, 13mpanl12 717 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 29 ≤ 𝑥) → (29↑2) ≤ (𝑥↑2))
155, 7, 14syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (ℤ29) → (29↑2) ≤ (𝑥↑2))
161nnrei 10989 . . . . . . . . . . . . . . . . . . . 20 𝑁 ∈ ℝ
1711resqcli 12905 . . . . . . . . . . . . . . . . . . . 20 (29↑2) ∈ ℝ
18 prmlem2.lt . . . . . . . . . . . . . . . . . . . . . 22 𝑁 < 841
1910nn0cni 11264 . . . . . . . . . . . . . . . . . . . . . . . 24 29 ∈ ℂ
2019sqvali 12899 . . . . . . . . . . . . . . . . . . . . . . 23 (29↑2) = (29 · 29)
21 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 29 = 29
22 1nn0 11268 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℕ0
23 6nn0 11273 . . . . . . . . . . . . . . . . . . . . . . . . 25 6 ∈ ℕ0
248, 23deccl 11472 . . . . . . . . . . . . . . . . . . . . . . . 24 26 ∈ ℕ0
25 5nn0 11272 . . . . . . . . . . . . . . . . . . . . . . . . 25 5 ∈ ℕ0
26 8nn0 11275 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℕ0
27192timesi 11107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (2 · 29) = (29 + 29)
28 2p2e4 11104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (2 + 2) = 4
2928oveq1i 6625 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 + 2) + 1) = (4 + 1)
30 4p1e5 11114 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (4 + 1) = 5
3129, 30eqtri 2643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2 + 2) + 1) = 5
32 9p9e18 11587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (9 + 9) = 18
338, 9, 8, 9, 21, 21, 31, 26, 32decaddc 11532 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (29 + 29) = 58
3427, 33eqtri 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 · 29) = 58
35 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 26 = 26
36 5p2e7 11125 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (5 + 2) = 7
3736oveq1i 6625 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((5 + 2) + 1) = (7 + 1)
38 7p1e8 11117 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (7 + 1) = 8
3937, 38eqtri 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((5 + 2) + 1) = 8
40 4nn0 11271 . . . . . . . . . . . . . . . . . . . . . . . . 25 4 ∈ ℕ0
41 8p6e14 11576 . . . . . . . . . . . . . . . . . . . . . . . . 25 (8 + 6) = 14
4225, 26, 8, 23, 34, 35, 39, 40, 41decaddc 11532 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2 · 29) + 26) = 84
43 9t2e18 11623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (9 · 2) = 18
44 1p1e2 11094 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 + 1) = 2
45 8p8e16 11578 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (8 + 8) = 16
4622, 26, 26, 43, 44, 23, 45decaddci 11540 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((9 · 2) + 8) = 26
47 9t9e81 11630 . . . . . . . . . . . . . . . . . . . . . . . . 25 (9 · 9) = 81
489, 8, 9, 21, 22, 26, 46, 47decmul2c 11549 . . . . . . . . . . . . . . . . . . . . . . . 24 (9 · 29) = 261
4910, 8, 9, 21, 22, 24, 42, 48decmul1c 11547 . . . . . . . . . . . . . . . . . . . . . . 23 (29 · 29) = 841
5020, 49eqtri 2643 . . . . . . . . . . . . . . . . . . . . . 22 (29↑2) = 841
5118, 50breqtrri 4650 . . . . . . . . . . . . . . . . . . . . 21 𝑁 < (29↑2)
52 ltletr 10089 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℝ ∧ (29↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (29↑2) ∧ (29↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2)))
5351, 52mpani 711 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℝ ∧ (29↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((29↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
5416, 17, 53mp3an12 1411 . . . . . . . . . . . . . . . . . . 19 ((𝑥↑2) ∈ ℝ → ((29↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
556, 15, 54sylc 65 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ29) → 𝑁 < (𝑥↑2))
56 ltnle 10077 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
5716, 6, 56sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ29) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
5855, 57mpbid 222 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℤ29) → ¬ (𝑥↑2) ≤ 𝑁)
5958pm2.21d 118 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℤ29) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥𝑁))
6059adantld 483 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ℤ29) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
6160adantl 482 . . . . . . . . . . . . . 14 ((¬ 2 ∥ 29 ∧ 𝑥 ∈ (ℤ29)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
62 9nn 11152 . . . . . . . . . . . . . . . 16 9 ∈ ℕ
63 3nn 11146 . . . . . . . . . . . . . . . 16 3 ∈ ℕ
64 1lt9 11189 . . . . . . . . . . . . . . . 16 1 < 9
65 1lt3 11156 . . . . . . . . . . . . . . . 16 1 < 3
66 9t3e27 11624 . . . . . . . . . . . . . . . 16 (9 · 3) = 27
6762, 63, 64, 65, 66nprmi 15345 . . . . . . . . . . . . . . 15 ¬ 27 ∈ ℙ
6867pm2.21i 116 . . . . . . . . . . . . . 14 (27 ∈ ℙ → ¬ 27 ∥ 𝑁)
69 7nn0 11274 . . . . . . . . . . . . . . 15 7 ∈ ℕ0
70 eqid 2621 . . . . . . . . . . . . . . 15 27 = 27
71 7p2e9 11132 . . . . . . . . . . . . . . 15 (7 + 2) = 9
728, 69, 8, 70, 71decaddi 11539 . . . . . . . . . . . . . 14 (27 + 2) = 29
7361, 68, 72prmlem0 15755 . . . . . . . . . . . . 13 ((¬ 2 ∥ 27 ∧ 𝑥 ∈ (ℤ27)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
74 5nn 11148 . . . . . . . . . . . . . . 15 5 ∈ ℕ
75 1lt5 11163 . . . . . . . . . . . . . . 15 1 < 5
76 5t5e25 11599 . . . . . . . . . . . . . . 15 (5 · 5) = 25
7774, 74, 75, 75, 76nprmi 15345 . . . . . . . . . . . . . 14 ¬ 25 ∈ ℙ
7877pm2.21i 116 . . . . . . . . . . . . 13 (25 ∈ ℙ → ¬ 25 ∥ 𝑁)
79 eqid 2621 . . . . . . . . . . . . . 14 25 = 25
808, 25, 8, 79, 36decaddi 11539 . . . . . . . . . . . . 13 (25 + 2) = 27
8173, 78, 80prmlem0 15755 . . . . . . . . . . . 12 ((¬ 2 ∥ 25 ∧ 𝑥 ∈ (ℤ25)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
82 prmlem2.23 . . . . . . . . . . . . 13 ¬ 23 ∥ 𝑁
8382a1i 11 . . . . . . . . . . . 12 (23 ∈ ℙ → ¬ 23 ∥ 𝑁)
84 3nn0 11270 . . . . . . . . . . . . 13 3 ∈ ℕ0
85 eqid 2621 . . . . . . . . . . . . 13 23 = 23
86 3p2e5 11120 . . . . . . . . . . . . 13 (3 + 2) = 5
878, 84, 8, 85, 86decaddi 11539 . . . . . . . . . . . 12 (23 + 2) = 25
8881, 83, 87prmlem0 15755 . . . . . . . . . . 11 ((¬ 2 ∥ 23 ∧ 𝑥 ∈ (ℤ23)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
89 7nn 11150 . . . . . . . . . . . . 13 7 ∈ ℕ
90 1lt7 11174 . . . . . . . . . . . . 13 1 < 7
91 7t3e21 11609 . . . . . . . . . . . . 13 (7 · 3) = 21
9289, 63, 90, 65, 91nprmi 15345 . . . . . . . . . . . 12 ¬ 21 ∈ ℙ
9392pm2.21i 116 . . . . . . . . . . 11 (21 ∈ ℙ → ¬ 21 ∥ 𝑁)
94 eqid 2621 . . . . . . . . . . . 12 21 = 21
95 1p2e3 11112 . . . . . . . . . . . 12 (1 + 2) = 3
968, 22, 8, 94, 95decaddi 11539 . . . . . . . . . . 11 (21 + 2) = 23
9788, 93, 96prmlem0 15755 . . . . . . . . . 10 ((¬ 2 ∥ 21 ∧ 𝑥 ∈ (ℤ21)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
98 prmlem2.19 . . . . . . . . . . 11 ¬ 19 ∥ 𝑁
9998a1i 11 . . . . . . . . . 10 (19 ∈ ℙ → ¬ 19 ∥ 𝑁)
100 eqid 2621 . . . . . . . . . . 11 19 = 19
101 9p2e11 11579 . . . . . . . . . . 11 (9 + 2) = 11
10222, 9, 8, 100, 44, 22, 101decaddci 11540 . . . . . . . . . 10 (19 + 2) = 21
10397, 99, 102prmlem0 15755 . . . . . . . . 9 ((¬ 2 ∥ 19 ∧ 𝑥 ∈ (ℤ19)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
104 prmlem2.17 . . . . . . . . . 10 ¬ 17 ∥ 𝑁
105104a1i 11 . . . . . . . . 9 (17 ∈ ℙ → ¬ 17 ∥ 𝑁)
106 eqid 2621 . . . . . . . . . 10 17 = 17
10722, 69, 8, 106, 71decaddi 11539 . . . . . . . . 9 (17 + 2) = 19
108103, 105, 107prmlem0 15755 . . . . . . . 8 ((¬ 2 ∥ 17 ∧ 𝑥 ∈ (ℤ17)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
109 5t3e15 11595 . . . . . . . . . 10 (5 · 3) = 15
11074, 63, 75, 65, 109nprmi 15345 . . . . . . . . 9 ¬ 15 ∈ ℙ
111110pm2.21i 116 . . . . . . . 8 (15 ∈ ℙ → ¬ 15 ∥ 𝑁)
112 eqid 2621 . . . . . . . . 9 15 = 15
11322, 25, 8, 112, 36decaddi 11539 . . . . . . . 8 (15 + 2) = 17
114108, 111, 113prmlem0 15755 . . . . . . 7 ((¬ 2 ∥ 15 ∧ 𝑥 ∈ (ℤ15)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
115 prmlem2.13 . . . . . . . 8 ¬ 13 ∥ 𝑁
116115a1i 11 . . . . . . 7 (13 ∈ ℙ → ¬ 13 ∥ 𝑁)
117 eqid 2621 . . . . . . . 8 13 = 13
11822, 84, 8, 117, 86decaddi 11539 . . . . . . 7 (13 + 2) = 15
119114, 116, 118prmlem0 15755 . . . . . 6 ((¬ 2 ∥ 13 ∧ 𝑥 ∈ (ℤ13)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
120 prmlem2.11 . . . . . . 7 ¬ 11 ∥ 𝑁
121120a1i 11 . . . . . 6 (11 ∈ ℙ → ¬ 11 ∥ 𝑁)
122 eqid 2621 . . . . . . 7 11 = 11
12322, 22, 8, 122, 95decaddi 11539 . . . . . 6 (11 + 2) = 13
124119, 121, 123prmlem0 15755 . . . . 5 ((¬ 2 ∥ 11 ∧ 𝑥 ∈ (ℤ11)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
125 9nprm 15762 . . . . . 6 ¬ 9 ∈ ℙ
126125pm2.21i 116 . . . . 5 (9 ∈ ℙ → ¬ 9 ∥ 𝑁)
127124, 126, 101prmlem0 15755 . . . 4 ((¬ 2 ∥ 9 ∧ 𝑥 ∈ (ℤ‘9)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
128 prmlem2.7 . . . . 5 ¬ 7 ∥ 𝑁
129128a1i 11 . . . 4 (7 ∈ ℙ → ¬ 7 ∥ 𝑁)
130127, 129, 71prmlem0 15755 . . 3 ((¬ 2 ∥ 7 ∧ 𝑥 ∈ (ℤ‘7)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
131 prmlem2.5 . . . 4 ¬ 5 ∥ 𝑁
132131a1i 11 . . 3 (5 ∈ ℙ → ¬ 5 ∥ 𝑁)
133130, 132, 36prmlem0 15755 . 2 ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
1341, 2, 3, 4, 133prmlem1a 15756 1 𝑁 ∈ ℙ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  cdif 3557  {csn 4155   class class class wbr 4623  cfv 5857  (class class class)co 6615  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901   < clt 10034  cle 10035  cn 10980  2c2 11030  3c3 11031  4c4 11032  5c5 11033  6c6 11034  7c7 11035  8c8 11036  9c9 11037  cdc 11453  cuz 11647  cexp 12816  cdvds 14926  cprime 15328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-rp 11793  df-fz 12285  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-dvds 14927  df-prm 15329
This theorem is referenced by:  37prm  15771  43prm  15772  83prm  15773  139prm  15774  163prm  15775  317prm  15776  631prm  15777  257prm  40802  139prmALT  40840  127prm  40844
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